However, far fewer results are known for variants of domination beyond trivial results such as for paths, cycles, stars, wheels, or complete graphs. We summarise the most noteworthy of these results for the variants of domination considered in this paper. For upper domination, results are known for various graphs based on chessboards [1]}, [2]}, [3]}, [4]}, [5]}. For total domination, results are known for some grid graphs \(G(n,m)\)  (for \(n \le 6\) ) [6]}, [7]}, Knödel graphs [8]}, and various graphs from chemistry [9]}. For connected domination, results are known for trees [10]}, circulant graphs [11]}, Centipede graphs [12]}, and various graphs from chemistry [9]}. For weak Roman domination, results are known for various graphs based on chessboards [14]}, Cartesian products involving complete graphs [15]}, and Helm graphs and Web graphs [16]}. For secure domination, results are known for some grid and torus graphs (for \(n \le 3\) ) [17]}, various Cartesian products of stars, cycles, paths and complete graphs [15]}, and middle graphs [19]}.
